The best way to calculate the logarithm and the inverse logarithm ($10^x$) is with a digital computer of some type, maybe a pocket calculator. For example, how do you find out that $\log_{10} 1.625 \approx 0.2108$? You put 1.625 into your calculator and push the log button; you calculate $10^{0.7469032}$ similarly.

Let's convert $2^{35}$ first :
$$2^{35}=10^{35\cdot \log_{10}(2)}\approx10^{35\cdot 0.30103}\approx 10^{10.53605}$$
$$\approx 10^{0.53605}\cdot 10^{10}\approx 3.436\cdot 10^{10}$$
We used a table of 'common logarithms' providing the decimals of $\log_{10}(x/100)$ with the results :

$\log_{10}(2)\approx 0.30103$ (see $x=200$) and

$\log_{10}(x/100)=0.536$ for $x$ between $343$ and $344$ getting $x/100=3.436$ by linear interpolation ($10^y$ is the inverse function of $\log_{10}(x)$)

The result is nearly :
$$-1.625\cdot 3.436\cdot 10^{10}\approx -5.5835\cdot 10^{10}$$

Of course most scientific calculators will give you directly the exact result : $$-1.625×2^{35}=-55834574848$$
You may verify this with a (reduced) table of powers of $2$ noticing that $-1.625=-\frac {13}8$ getting :
$$-\frac {13}{2^3}2^{35}=-13\cdot 2^{32}=-13\cdot 4\cdot\bigl(2^{10}\bigr)^3=-52\cdot1024^3$$